Resolving an \( A_2 \) singularity, also known as a simple surface singularity or Du Val singularity of type \( A_2 \), involves a process in algebraic geometry where the singular point on a surface is replaced with a more regular structure, typically a smooth surface or a less singular variety. The \( A_2 \) singularity specifically refers to the singularity at the origin of the hypersurface given by the equation:

\[
x^2 + y^2 + z^3 = 0
\]

This is a type of rational double point, and it can be resolved by a procedure known as *blowing up*. Here’s a step-by-step outline of the process:

### Step 1: Understand the Nature of the Singularity
- The singularity \( A_2 \) corresponds to a point where the surface is not smooth but has a cusp-like behavior. It is a two-dimensional singularity.

### Step 2: Perform the Blow-Up
- **Blowing up** at the singular point introduces a new curve called the *exceptional divisor* which replaces the singular point with a smoother structure. This process typically involves replacing the singular point with a projective line \( \mathbb{P}^1 \).

### Step 3: Analyze the Exceptional Divisor
- After the blow-up, the exceptional divisor is usually a curve, and in the case of the \( A_2 \) singularity, it corresponds to a chain of 2 rational curves (i.e., \( \mathbb{P}^1 \)) intersecting transversely at a point.

### Step 4: Check for Smoothness
- The surface after blowing up should be checked to ensure that the singularity is resolved, meaning the resulting surface is smooth. In the case of an \( A_2 \) singularity, the process will indeed result in a smooth surface.

### Step 5: Interpret the Result
- The resulting space after resolving the singularity can be interpreted as having replaced the original singular point with a small chain of projective lines, leading to a more regular geometry.

### Summary
- An \( A_2 \) singularity is resolved by blowing up the singular point, introducing a chain of rational curves in place of the singularity, which results in a smooth surface.

Would you like to explore further details on any of these steps?

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Here are some related questions to expand your understanding:

1. What is the geometric interpretation of the blow-up process?
2. How does the resolution of singularities differ between \( A_n \) types (e.g., \( A_1 \), \( A_2 \), \( A_3 \))?
3. What are the implications of resolving singularities in complex surfaces?
4. Can you describe the role of exceptional divisors in resolving singularities?
5. What are the differences between resolving singularities in algebraic curves versus surfaces?

**Tip:** When working with singularities in algebraic geometry, understanding the local structure around the singularity is crucial, as it guides the resolution process.

Explore how to resolve an A_2 singularity in algebraic geometry with a detailed step-by-step guide. Understand the blow-up process, exceptional divisors, and the transformation from a singular point to a smooth surface. Suitable for advanced mathematics enthusiasts.

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